Inner mantles and iterated HOD

Mathematical Logic Quarterly 65 (4):498-510 (2019)
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Abstract

We present a class forcing notion, uniformly definable for ordinals η, which forces the ground model to be the ηth inner mantle of the extension, in which the sequence of inner mantles has length at least η. This answers a conjecture of Fuchs, Hamkins, and Reitz [1] in the positive. We also show that forces the ground model to be the ηth iterated of the extension, where the sequence of iterated s has length at least η. We conclude by showing that the lengths of the sequences of inner mantles and of iterated s can be separated to be any two ordinals you please.

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Citations of this work

Cohen forcing and inner models.Jonas Reitz - 2020 - Mathematical Logic Quarterly 66 (1):65-72.

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References found in this work

Classes and truths in set theory.Kentaro Fujimoto - 2012 - Annals of Pure and Applied Logic 163 (11):1484-1523.
Set-theoretic geology.Gunter Fuchs, Joel David Hamkins & Jonas Reitz - 2015 - Annals of Pure and Applied Logic 166 (4):464-501.
The downward directed grounds hypothesis and very large cardinals.Toshimichi Usuba - 2017 - Journal of Mathematical Logic 17 (2):1750009.
Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
The Ground Axiom.Jonas Reitz - 2007 - Journal of Symbolic Logic 72 (4):1299 - 1317.

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